Reaction-diffusion Cellular Neural Network Research Articles

Boundary stabilisation of fractional reaction-diffusion systems with time-varying delays

This paper examines the boundary stabilisation results for time fractional-order reaction-diffusion systems involving with time-varying delays. The main goal is to design the boundary control for the system by proving the well-posedness of the kernel function using the backstepping method. An invertible Volterra integral transformation is used to convert the considered system into a stable target system. Different from the existing results, the stability results for fractional RDEs are discussed in the sense of the Lyapunov–Krasovskii theory and sufficient conditions are derived with the help of the linear matrix inequality (LMI) approach. Finally, to show the application of the results, the proposed conditions are numerically validated over a time fractional-order reaction-diffusion cellular neural network model.

Edge of Chaos Explains Prigogine’s Instability of the Homogeneous

One of the complex phenomena, which most attracted the attention of the scientific community over the past few decades, is the emergence of diffusion-driven instabilities in homogeneous cellular media. Explaining this symmetry-breaking process, which the Russian luminary Ilya Prigogine referred to as the Instability of the Homogeneous, is only possible upon invoking the Physics Law of the Edge of Chaos. While the emergence of inhomogeneous static or dynamic periodic solutions in biological reaction-diffusion systems was recently reproduced in two-cell networks, leveraging the Negative Differential Resistance of certain volatile memristors, so far, to the best of our knowledge, no cellular array was ever found to undergo dissipation-induced dynamical phenomena of higher complexity, including a route to chaos, or an uncontrollable upsurge in the respective physical variables. This manuscript determines all the possible sets of conditions on the parameters of a one-port, extracted from the classical Chua circuit, enabling to poise it on some stable and locally-active operating point, employing a deep mathematical analysis, focusing on the properties of its local impedance. The polarisation of the one-port on some Edge of Chaos operating point is the Conditio Sine Qua Non for the Destabilisation of the Homogeneous Solution, resulting in the steady-state appearance of high-order dynamical phenomena, as outlined above, in a simple Reaction-Diffusion Cellular Nonlinear Network, where a linear and passive resistor, interposed between two identical copies of the one-port, lets them interact by means of a diffusion process. Notably, as demonstrated here, whether or not the dissipative coupling path may truly succeed in moving each of the two one-ports of the two-cell array, under study, away from the respective common Edge of Chaos operating point, depends critically upon the satisfaction of the logical disjunction between three inequalities, which enforces the instability of the denominator polynomial of the rational function associated to the local input impedance of the cellular array, as seen across either of the two one-ports.

Boundary intermittent stabilization for delay reaction–diffusion cellular neural networks

Boundary intermittent stabilization for delay reaction–diffusion cellular neural networks

EDGE DETECTION ALGORITHM FOR COLOR IMAGES BASED ON THE REACTION-DIFFUSION EQUATION AND THE CELLULAR NEURAL NETWORK MODEL

EDGE DETECTION ALGORITHM FOR COLOR IMAGES BASED ON THE REACTION-DIFFUSION EQUATION AND THE CELLULAR NEURAL NETWORK MODEL

Boundary Mittag-Leffler stabilization of fractional reaction–diffusion cellular neural networks

Mittag-Leffler stabilization is studied for fractional reaction–diffusion cellular neural networks (FRDCNNs) in this paper. Different from previous literature, the FRDCNNs in this paper are high-dimensional systems, and boundary control and observed-based boundary control are both used to make FRDCNNs achieve Mittag-Leffler stability. First, a state-dependent boundary controller is designed when system states are available. By employing the spatial integral functional method and some inequalities, a criterion ensuring Mittag-Leffler stability of FRDCNNs is presented. Then, when the information of system states is not fully accessible, an observer is presented to estimate the system states based on boundary output and an observer-based boundary controller is provided aiming to stabilize the considered FRDCNNs. Furthermore, a robust observer-based boundary controller is proposed to ensure the Mittag-Leffler stability for FRDCNNs with uncertainties. Examples are given to illustrate the effectiveness of obtained theoretical results.

Delayed Reaction–Diffusion Cellular Neural Networks of Fractional Order: Mittag–Leffler Stability and Synchronization

This research introduces a model of a delayed reaction–diffusion fractional neural network with time-varying delays. The Mittag–Leffler-type stability of the solutions is investigated, and new sufficient conditions are established by the use of the fractional Lyapunov method. Mittag–Leffler-type synchronization criteria are also derived. Three illustrative examples are established to exhibit the proposed sufficient conditions.

Edge of chaos in reaction diffusion CNN model

Abstract In this paper, we study the dynamics of a reaction-diffusion Cellular Nonlinear Network (RD-CNN) nodel in which the reaction term is represented by Brusselator cell. We investigate the RD-CNN dynamics by means of describing function method. Comparison with classical results for Brusselator equation is provided. Then we introduce a new RD-CNN model with memristor coupling, for which the edge of chaos regime in the parameter space is determined. Numerical simulations are presented for obtaining dynamic patterns in the RD-CNN model with memristor coupling.

Stability Analysis of Impulsive Stochastic Reaction-Diffusion Cellular Neural Network with Distributed Delay via Fixed Point Theory

This paper investigates the stochastically exponential stability of reaction-diffusion impulsive stochastic cellular neural networks (CNN). The reaction-diffusion pulse stochastic system model characterizes the complexity of practical engineering and brings about mathematical difficulties, too. However, the difficulties have been overcome by constructing a new contraction mapping and an appropriate distance on a product space which is guaranteed to be a complete space. This is the first time to employ the fixed point theorem to derive the stability criterion of reaction-diffusion impulsive stochastic CNN with distributed time delays. Finally, an example is provided to illustrate the effectiveness of the proposed methods.

Global μ-stability of impulsive reaction–diffusion neural networks with unbounded time-varying delays and bounded continuously distributed delays

This paper investigates the global μ-stability for impulsive cellular neural networks with reaction–diffusion terms and mixed delays, where mixed delays consist of unbounded time-varying delays and bounded continuously distributed delays. The model under consideration is more general than those investigated in most existing literature. By establishing a novel Lyapunov–Krasovskii functional and employing the delay differential inequality with impulsive initial conditions, boundary conditions and LMI, we obtain two concise and sufficient conditions ensuring global μ-stability of the impulsive reaction–diffusion cellular neural networks with mixed delays. The newly obtained criteria are concerned with the reaction–diffusion coefficients and the regional feature. However, they are independent of boundaries of variable time delays. Several corollaries are also presented. Finally, three concrete numerical examples are given to demonstrate the effectiveness and superiority of our main results.

Exponential Stability and Periodicity of Fuzzy Delayed Reaction-Diffusion Cellular Neural Networks with Impulsive Effect

This paper considers dynamical behaviors of a class of fuzzy impulsive reaction-diffusion delayed cellular neural networks (FIRDDCNNs) with time-varying periodic self-inhibitions, interconnection weights, and inputs. By using delay differential inequality,M-matrix theory, and analytic methods, some new sufficient conditions ensuring global exponential stability of the periodic FIRDDCNN model with Neumann boundary conditions are established, and the exponential convergence rate index is estimated. The differentiability of the time-varying delays is not needed. An example is presented to demonstrate the efficiency and effectiveness of the obtained results.

Global exponential synchronization of generalized stochastic neural networks with mixed time-varying delays and reaction-diffusion terms

This paper investigates the synchronization problem of generalized stochastic neural networks with mixed time-varying delays and reaction-diffusion terms using linear feedback control. Lyapunov stability theory combining with stochastic analysis approaches is employed to derive sufficient criteria ensuring the coupled chaotic generalized stochastic neural networks to be globally exponentially synchronized. The generalized neural networks model considered includes reaction-diffusion Hopfield neural networks, reaction-diffusion bidirectional associative memory neural networks, and reaction-diffusion cellular neural networks as its special cases. It is theoretically proven that these synchronization criteria are more effective than some existing ones. This paper also presents some illustrative examples and uses simulated results of these examples to show the feasibility and effectiveness of the proposed scheme.

Theorems and application of local activity of CNN with five state variables and one port.

Coupled nonlinear dynamical systems have been widely studied recently. However, the dynamical properties of these systems are difficult to deal with. The local activity of cellular neural network (CNN) has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice, which is composed of coupled cells. In this paper, the analytical criteria for the local activity in reaction-diffusion CNN with five state variables and one port are presented, which consists of four theorems, including a serial of inequalities involving CNN parameters. These theorems can be used for calculating the bifurcation diagram to determine or analyze the emergence of complex dynamic patterns, such as chaos. As a case study, a reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is analyzed and simulated, the bifurcation diagram is calculated. Using the diagram, numerical simulations of this CNN model provide reasonable explanations of complex mutant phenomena during therapy. Therefore, it is demonstrated that the local activity of CNN provides a practical tool for the complex dynamics study of some coupled nonlinear systems.

The Globally Asymptotic Stability of Reaction-Diffusion Cellular Neural Networks with Time Delays

A new method is presented to investigate the globally asymptotic stability of reaction-diffusion cellular neural network with time-delays. The method is built on the mean Lyapunov function. We give some new sufficient conditions on time-dependent or time-independent to ensure globally asymptotic stability of the equilibrium point of the class of systems under the conditions of the active functions are continuous by employing linear matrix inequality (LMI). The results extend and improve the earlier works of other researchers. In additions, examples are given to show the application of the established stability theorems.

GLOBAL EXPONENTIAL STABILITY OF CHUA'S REACTION–DIFFUSION CNN

In this paper, the global exponential stability of Chua's reaction–diffusion CNN system is investigated. For this system, some sufficient conditions ensuring the existence and global exponential stability of the equilibrium point is derived by using homeomorphism mapping, the property of coefficient matrix and analytical techniques. Finally, three illustrative examples are given to show the effectiveness of our results.

Global exponential synchronization in delayed reaction-diffusion cellular neural networks with the Dirichlet boundary conditions

The global exponential synchronization for a class of delayed reaction-diffusion cellular neural networks with Dirichlet boundary conditions is discussed. Some new sufficient conditions which include the diffusion coefficients are obtained by using the Lyapunov functional method, many real parameters and inequality techniques. Particularly, different from previous works, see references [S. Li, H. Yang, X. Lou, Adaptive exponential synchronization of delayed neural networks with reaction-diffusion terms, Chaos, Solitons Fractals 40 (2009) 930–939; X. Lou, B. Cui, Asymptotic synchronization of a class of neural networks with reaction-diffusion terms and time-varying delayes, Comput. Math. Appl. 52 (2006) 897–904; Y. Wang, J. Cao, Synchronization of a class of delayed neural networks with reaction–diffusion terms, Phys. Lett. A 369 (2007) 201–211], in our results the effect of the diffusion terms on the synchronization are considered for the first time. Finally, a numerical example is given to verify our results.

Cellular Nonlinear Networks for the emergence of perceptual states: Application to robot navigation control

In this paper a new general purpose perceptual control architecture, based on nonlinear neural lattices, is presented and applied to solve robot navigation tasks. Insects show the ability to react to certain stimuli with simple reflexes, using direct sensory-motor pathways, which can be considered as basic behaviors, inherited and pre-wired. Relevant brain centres, known as Mushroom Bodies (MB) and Central Complex (CX) were recently identified in insects: though their functional details are not yet fully understood, it is known that they provide secondary pathways allowing the emergence of cognitive behaviors. These are gained through the coordination of the basic abilities to satisfy the insect's needs. Taking inspiration from this evidence, our architecture modulates, through a reinforcement learning, a set of competitive and concurrent basic behaviors in order to accomplish the task assigned through a reward function. The core of the architecture is constituted by the so-called Representation layer, used to create a concise picture of the current environment situation, fusing together different stimuli for the emergence of perceptual states. These perceptual states are steady state solutions of lattices of Reaction-Diffusion Cellular Nonlinear Networks (RD-CNN), designed to show Turing patterns. The exploitation of the dynamics of the multiple equilibria of the network is emphasized through the adaptive shaping of the basins of attraction for each emerged pattern. New experimental campaigns on standard robotic platforms are reported to demonstrate the potentiality and the effectiveness of the approach.

A Simple Programmable Autowave Generator Network for Wave Computing Applications

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Spatiotemporal-wave-based computing on an active medium, which is known as wave computing paradigm, mimics the way biological systems process information. Recent anatomic and physiological studies and technological developments in very large-scale integration technology have encouraged the engineers to brace up with this new paradigm. In this paper, a simple network model that is essential in order to explore spatiotemporal behavior and to develop new algorithms based on wave computing technique has been presented. The presented network is a 2-D reaction-diffusion cellular neural network consisting of relaxation oscillators. The network can be programmed as an autowave generator and sources of the wave can be placed at any place on the network. The network has the simplest model that exhibits autowaves. Furthermore, the model is suitable for the implementation of a larger network. The propagation of autowaves between the cells whose dynamics is fixed shows that the network can be adapted to already developed wave computing algorithms in literature. </para>

W 1,2(Ω)- and X 1,2(Ω)-stability of reaction-diffusion cellular neural networks with delay

With Poincare’s inequality and auxiliary function applied in a class of retarded cellular neural networks with reaction-diffusion, the conditions of the systems’ W1,2(Ω)-exponential and X1,2(Ω)-asmptotic stability are obtained. The stability conditions containing diffusion term are different from those obtained in the previous papers in their exponential stability conditions. One example is given to illustrate the feasibility of this method in the end.

Perception for Action: Dynamic Spatiotemporal Patterns Applied on a Roving Robot

In this article, we apply a bio-inspired control architecture to a roving robot performing different tasks. The key of the control system is the perceptual core, where heterogeneous information coming from sensors is merged to build an internal portrait representing the current situation of the environment. The internal representation triggers an action as the response to the current stimuli, closing the loop between the agent and the external world. The robot's internal state is implemented through a nonlinear lattice of neuron cells, allowing the generation of a large amount of emergent steady-state solutions in the form of Turing patterns. These are incrementally shaped, through learning, so as to constitute a “mirror” of the environmental conditions. Reaction—diffusion cellular nonlinear networks were chosen to generate Turing patterns as internal representations of the robot surroundings. The associations between incoming sensations and the perceptual core, and between Turing patterns and actions to be performed, are driven by two reward-based learning mechanisms. We report on simulation results and experiments on a roving robot to show the suitability of the approach.

Global exponential stability of reaction-diffusion cellular neural networks with S-type distributed time delays

The authors discuss the existence of the equilibrium point and its global exponential stability of reaction-diffusion cellular neural networks (RDCNNs) with S-type distributed signal transmission delays along the axon of a neuron by means of the topological degree theory and differential inequality technique. A theorem and two corollaries were obtained in which the boundedness, monotonicity and differentiability conditions on the activation functions are not required. The sufficient conditions on global exponential stability established in this paper, which are easily verifiable, have a wider adaptive range.